Let \(F\) be a field of characteristic zero. The study of associative algebras over \(F\) satisfying polynomial identities reduces then to study multilinear polynomial identities. Let \(P_n\) be the linear span of all multilinear polynomials in \(x_1,\ldots,x_n\) (i.e. the span of \(\{x_{\sigma(1)},\ldots,x_{\sigma(n)}\mid \sigma\in S_n\}\)) in the free associative algebra \(F(X)\) for an infinite countable set \(X\). The \(n\)-th codimension \(c_n(A)\) of an associative algebra \(A\) is defined to be the dimension of the quotient \(P_n(A)=P_n/(P_n\cap T(A))\), where \(T(A)\) denotes the T-ideal of \(A\). Codimensions of an associative algebra give rise to a convergent sequence \((c_n(A))^{1/n}\) whose limit, always integer, is called the PI-exponent of \(A\).
If one assumes the associative algebra \(A\) to be graded by a group \(G\) and considers graded (multilinear) polynomials in the free associative \(G\)-graded algebra \(F(X,G)\), the ideal \(\text{Id}^G(A)\) of graded identities of \(A\) is determined by its multilinear polynomials, so that by the subspaces \(P_n^G\cap \text{Id}^G(A)\), \(n\geq1\), where \(P_n^G\) denotes the space of multilinear \(G\)-graded polynomials of degree \(n\).
For any decomposition \(n=n_1+\cdots+n_s\) of \(n\), the subspace \(P_n\) is a direct sum of subspaces \(P_{n_1,\ldots,n_s}\), where \(P_{n_1,\ldots,n_s}\) denotes the space of multilinear graded polynomials with \(n_i\) variables of homogeneous degree \(g_i\), \(i=1,\ldots,s\). Then \(S_{n_1}\times\cdots\times S_{n_s}\) acts on \(P_{n_1,\ldots,n_s}(A)=P_{n_1,\ldots,n_s}/(P_{n_1,\ldots,n_s}\cap \text{Id}^G(A))\) by permuting the variables of the same homogeneous degree, i.e. \(S_{n_i}\) permutes the variables of homogeneous degree \(g_i\), \(i=1,\ldots,s\), endowing \(P_{n_1,\ldots,n_s}(A)\) with a \(S_{n_1}\times\cdots\times S_{n_s}\)-module structure having a completely irreducible character \( \chi^G_{n_1,\ldots,n_s}(A)\) which is called to be the \( (n_1,\ldots,n_s) \)-th cocharacter of \(A\). Writing the \( (n_1,\ldots,n_s) \)-th cocharacter as a sum of irreducible characters, multiplicities of the irreducible components indicate the maximal number of linearly independent highest weight vectors.
Analogous notions can be considered for Jordan algebras. Moreover, in the case of unitary algebras, it suffices to consider multilinear proper polynomials (i.e. linear combinations of products of commutators, of length at least two, in the free associative algebra). This paper focus on the Jordan algebra \(UJ_2=UJ_2(F)\) of \(2\times 2\) upper triangular matrices over a field \(F\) of characteristic zero. \( \mathbb{Z}_2\)-gradings on \(UJ_2 \) where classified up to isomorphism by P. Koshlukov and F. Martino [J. Pure Appl. Algebra 216, No. 11, 2524–2532 (2012; Zbl 1287.17053)] where the generators of the \(T_2\)-ideal of graded identities for the different \( \mathbb{Z}_2\)-gradings on \(UJ_2 \) were also computed.
Now, for each \( \mathbb{Z}_2\)-grading the \( \mathbb{Z}_2\)-graded cocharacter sequence of \(UJ_2^{\mathbb{Z}_2}\) and their multiplicities are computed. Finally the non-graded case is considered to compute the ordinary \(n\)th cocharacters of \(UJ_2(F) \), for any infinite field \(F\) of characteristic different from 2 and 3.